A relative of mine was killed by a drunk driver in 1987 in California.
 I was only 7 years old at the time so I was not told what happened to
the driver who killed him.  No one in my family seems to know what
happened to the driver (whether he/she was killed, served jail time,
got away with it).  Is there any way to search some kind of public
record database to find out what happened to the drunk driver's case?  I know my
relative's whole first and last name, date of birth, date of death,
and that he lived in (or around) Los Angeles.  Where can I go from here?

I would start with the police department in the area in which the
fatality occured. The driver was most certainly cited for 'something',
and there will be a record of it. There will be an accident report. I
don't know exactly how to find it, but polite persistance will get you
nearly anything. These records are public, but in most instances you
will need to explain your reason for the request. Once you have the
accident report, the name of the person you wish to investigate will
be listed. I would then call the Prosecutor's Office and pray you get
a friendly soul that will assist you by looking back for you to see
what became of the ticket issued to the drunk driver. Of course if the
drunk driver died as well then nothing further would have come of it,
unless that person did not OWN the vehicle they were driving.

If you choose to follow the approach suggested in that comment, and if
the accident occurred in Los Angeles County, the best point of contact
for the Los Angeles County District Attorney's Office is probably the
Victim-Witness Assistance Program:
http://da.co.la.ca.us/vwap/default.htm

Los Angeles County District Attorney's Office
Victim-Witness Assistance Program
210 W. Temple Street, Room 12-514
Los Angeles, CA 90012
Phone: (213) 974-7499
1-800-773-7574